Question

In a $\Delta ABC$, if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A.\sin B.\sin C=\frac{3+\sqrt{3}}{8}$,

then- On the basis of above information, answer the following questions:The value of $\tan A+\tan B+\tan C$ is:

• A. $\frac{3+\sqrt{3}}{\sqrt{3}-1}$
• B. $\frac{\sqrt{3}+4}{\sqrt{3}-1}$
• C. $\frac{6-\sqrt{3}}{\sqrt{3}-1}$
• D. $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-1}$

Answer 1

Answer: (A)

$\frac{3+\sqrt{3}}{\sqrt{3}-1}$

We have, $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A\sin B\sin C=\frac{3+\sqrt{3}}{8}$

$\Rightarrow \tan A\tan B\tan C=\frac{3+\sqrt{3}}{\sqrt{3}-1}......\left(i\right)$

But, $\tan A+\tan B+\tan C=\tan A.\tan B.\tan C$ (using conditional identity)

$\tan A+\tan B+\tan C=\frac{3+\sqrt{3}}{\sqrt{3}-1}$.....(ii)